/// <summary>
/// Performs one iteration of the secant method on the <paramref name="bracket"/> interval.
/// </summary>
/// <param name="bracket">The bracket.</param>
/// <param name="values">The values.</param>
/// <returns>System.Double.</returns>
private double Secant(Bracket bracket, ref FunctionValues values)
{
var a = bracket.Start;
var b = bracket.End;
var dφa = values.dφ(a);
var dφb = values.dφ(b);
var c = (a * dφb - b * dφa) / (dφb - dφa);
return(c);
}
/// <summary>
/// Determines if the algorithm should terminate, given the currently selected step width <paramref name="α" />
/// and the shared <paramref name="values"/>.
/// </summary>
/// <param name="values">The values.</param>
/// <param name="α">The selected step width.</param>
/// <returns><see langword="true" /> if the line search should terminate, <see langword="false" /> otherwise.</returns>
private bool ShouldTerminate(double α, ref FunctionValues values)
{
// calculate the function values at α
var φα = values.φ(α);
var dφα = values.dφ(α);
// delegate
return(ShouldTerminate(α, values.φ0, values.dφ0, φα, dφα));
}
/// <summary>
/// Updates the <paramref name="current" /> bracketing interval around the midpoint <paramref name="c" />.
/// </summary>
/// <param name="current">The current bracketing interval.</param>
/// <param name="c">The midpoint.</param>
/// <param name="values">The values.</param>
/// <returns>Bracket.</returns>
/// <exception cref="System.NotImplementedException"></exception>
private Bracket UpdateBracketing(Bracket current, double c, ref FunctionValues values)
{
// we do not check for infinity here, as this will be handled in U0
Debug.Assert(!Double.IsNaN(c), "!Double.IsNaN(c)");
// prefetch
var θ = _θ; // theta
var ε = _ε; // epsilon
// helpers
var a = current.Start;
var b = current.End;
// U0: if the midpoint is out of range of the current bracketing
// interval, stay where we are.
if (c < a || c > b)
{
return(current);
}
// U1
var dφc = values.dφ(c);
if (dφc >= 0.0D)
{
return(new Bracket(start: a, end: c));
}
// U2
var φc = values.φ(c);
if (φc <= values.φ0 + ε)
{
return(new Bracket(start: c, end: b));
}
// U3
return(UpdateBracketInRange(start: a, end: c, values: ref values));
}
/// <summary>
/// Updates the bracketing in range <see cref="start" /> to <see cref="end" />.
/// </summary>
/// <param name="start">The starting point.</param>
/// <param name="end">The end point.</param>
/// <param name="values">The values.</param>
/// <returns>Bracket.</returns>
private Bracket UpdateBracketInRange(double start, double end, ref FunctionValues values)
{
Debug.Assert(start.IsFinite(), "start.IsFinite()");
Debug.Assert(end.IsFinite(), "end.IsFinite()");
// prefetch
var θ = _θ; // theta
var ɛ = _ε; // epsilon
var φ0 = values.φ0;
var currentStart = start;
var currentEnd = end;
var previousD = double.NaN;
while (true) // TODO: bug in disguise?
{
// U3a
var d = (1 - θ) * currentStart + θ * currentEnd;
Debug.Assert(d != previousD, "d != previousD"); // NOTE that this case is not in the paper
previousD = d;
if (values.dφ(d) >= 0.0D)
{
return(new Bracket(start: currentStart, end: d));
}
// U3b: close in from the left
if (values.φ(d) <= φ0 + ɛ)
{
currentStart = d;
continue;
}
// U3c: close in from the right
currentEnd = d;
}
}
/// <summary>
/// Brackets the starting point <paramref name="α"/>.
/// </summary>
/// <param name="α">The starting point.</param>
/// <param name="values">The values.</param>
/// <returns>Bracket.</returns>
/// <exception cref="System.NotImplementedException"></exception>
private Bracket BracketStartingPoint(double α, FunctionValues values)
{
// prefetch
var ρ = _ρ; // rho
var θ = _θ; // theta
var ε = _ε; // epsilon
// make sure the parameters are in range
Debug.Assert(ρ > 1, "ρ > 1");
Debug.Assert(θ > 0 && θ < 1, "θ > 0 && θ < 1");
Debug.Assert(α >= 0, "α >= 0");
// B0: initialize the current step value cj and keep track of the values in a stack
var c = α;
var previousC = new Stack <double>();
var φ0 = values.φ0;
// TODO: set a maximum iteration count
var maxBracketingIterations = _maxBracketingIterations;
for (var j = 0; j < maxBracketingIterations; ++j)
{
// register the current step value
previousC.Push(c);
// determine the gradient at the current step length
var dφc = values.dφ(c);
// B1: if we find our currently selected end value to be ascending,
// then backtrack the starting value to the last descend.
// Note that we may end up here in the first iteration also
// if the start point generation yields a problematic result.
if (dφc >= 0)
{
var end = c;
// find the most recent step selection c smaller cj that resulted
// in a function value less than the starting point.
// Since we just added cj, we'll just throw that away.
previousC.Pop();
while (previousC.Count > 0)
{
c = previousC.Pop();
if (values.φ(c) > φ0 + ε)
{
continue;
}
var start = c;
return(new Bracket(start, end));
}
// at this point there was no good starting point.
// sadly, this point is not handled in the paper, so
// we'll just throw out 0 as the starting point and hope for the best.
return(new Bracket(0, end));
}
// B2: If, for some reason, we are on a descending direction, yet the
// current function value is larger than what we started with, then
// we know a minimum must exist between the current point and the start.
// By using the secant method, we zoom in the range until we find a valid
// search region.
if (values.φ(c) > φ0 + ε)
{
return(UpdateBracketInRange(start: 0.0D, end: c, values: ref values));
}
// B3: At this point, we are still descending and we did not skip
// any minima as far as we know, so we'll increase our step size.
c = ρ * c;
}
// welp
return(new Bracket(0, c));
}
